why do you even care?
seriously, the bottom of your ski rides on the snow. you cant even see it.
my bases have huge black streaks, im pretty sure it was from this greasy tire jib. but i live with it.
the best way to get rid of this is to just not give a fuck, and keep skiing. eventually, after enough shredding, the shit in your base will slowly seep out of the pores, leaving it mostly clean.
if you must have it out now, try to find some base cleaner, ask your ski shop for it, it is extremely bad for your base (it cleans out all the good shit, like wax, out too, and leaves your base dry as fuck ), but i guess thats what it takes to keep your skis looking good right?
or, you could wax your skis, the wax going into the pores of your base naturally pushes out the dirt and shit and cleans it a little.
but seriously i suggest chilling the fuck out.